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9

Fracture Patterns Induced by Desiccation in a Thin Layer

So Kitsunezaki∗

Department of Physics, Nara Women’s University, Nara 630-8506, Japan

(September 29, 2018)

Abstract

We study a theoretical model of mud cracks, that is, the fracture patterns

resulting from the contraction with drying in a thin layer of a mixture of

granules and water. In this model, we consider the slip on the bottom of this

layer and the relaxation of the elastic field that represents deformation of the

layer. Analysis of the one-dimensional model gives results for the crack size

that are consistent with experiments. We propose an analytical method of

estimation for the growth velocity of a simple straight crack to explain the

very slow propagation observed in actual experiments. Numerical simulations

reveal the dependence of qualitative nature of the formation of crack patterns

on material properties.

46.35.+z,46.50.+a,47.54.+r,62.20.Mk

Typeset using REVTEX

1kitsune@minnie.disney.phys.nara-wu.ac.jp

1

I. INTRODUCTION

Many kinds of mixtures of granules and water, such as clay, contract upon desiccation

and form cracks. These fracture patterns are familiar to us as ordinary mud cracks. However,

the fundamental questions about these phenomena have not yet been answered theoretically.

The problems which need to be addressed include determining the condition under which

fragmentation occurs, the dynamics displayed by cracks, and the patterns which grow.

In simple and traditional experiments on mud cracks, a thin layer of a mixture in a

rigid container with a horizontal bottom is prepared left to dry at room temperature [1–5].

Typically, clay, soil, flour, granules of magnesium carbonate and alumina are used. In

almost all cases, cracks extend from the surface to the bottom of the layer and propagate

horizontally along a line, forming a quasi-two-dimensional structure. Typically we observe

a tiling pattern composed of rectangular cells in which cracks mainly join in a T-shape.

Groisman and Kaplan carried out more detail experiments with coffee powder and reported

1) that the size of a crack cell after full drying is nearly proportional to the thickness of

the layer and larger in the case of a “slippery” bottom, 2) that the velocity of a moving

crack is almost independent of time for a given crack and very slow on the order of several

millimeters per minute, but that it differs widely from one crack to another, and 3) that as

the layer becomes thin, there is a transition to patterns which contain many Y-shape joints

and unclosed cells owing to the arrest of cracks [2].

Another experimental setup was used by Allain and Limat [6]. This setup produces cracks

that grow directionally by causing evaporation to proceed from one side of the container.

Sasa and Komatsu have proposed a theoretical model for such systems [7].

Fragmentation of coating or painting also arises from desiccation, This has been studied

theoretically by some people [8–10]. From the viewpoint that fractures are caused by slow

contraction, these problems can be thought of as belonging to the same category as thermal

cracks in glasses [11–16] and the formation of joints in rocks brought by cooling [17,18].

In addition, we note that mixtures of granular matter and fluid have properties that vary

2

greatly from that of complete elastic materials, in particular, dissipation and viscoelasticity.

The propagation of cracks in such media has been investigated recently using developments

in nonlinear physics [19–28].

In this paper, we undertake a theoretical investigation of the experiments described

above. We treat such system as consisting of fractures arising from quasi-static and uniform

contraction in thin layers of linear elastic material.

In Sec. II, we propose a one-dimensional model. Our model takes into account the slip

displacement on the bottom of a container, because most of the experiments can not be

assumed to obey a fixed boundary condition. We can investigate the development of the

size of a crack cell by applying a fragmentation condition to the balanced states of the elastic

field.

In Sec. III, we report the analytical results of our one-dimensional model. Here we

consider both the critical stress condition and the Griffith criterion as the fragmentation

condition. We consider these two alternative criteria because the nature of the breaking

condition in mixtures of granules and water is not clear. The critical stress condition predicts

that the final size of a crack cell is proportional to the thickness of the layer and that, in

the case of a slippery bottom, it becomes much larger than the thickness. These predictions

seem to be consistent with the experimental results. In contrast, we find that the Griffith

criterion predicts a different relation between the final size of a cell and the thickness.

In Sec. IV, we extend the model to two dimensions and investigate the time development

of a crack. In order to describe the relaxation process of the elastic field, we use the Kelvin

model while taking into account the effect of the bottom of the container. We assume the

stress, excluding dissipative force, to be constant in the front of a propagating crack tip and

evaluate the velocity of a simple straight crack tip analytically. Our results indicate that

cracks advance at very slow speed in comparison with the sound velocity.

In Sec. V, we report on the numerical simulations of our model that reproduce fracture

patterns similar to those in real experiments. The growth of the patterns exhibits qualitative

3

differences depending on the elastic constants and the relaxation time. As the relaxation

time becomes smaller, in particular, we observe the growth of fingering patterns with tip

splitting rather than side branching of cracks.

Finally, we conclude the paper with a summary of the results and a discussion of the

open problems in Sec. VI.

II. MODELING OF FRACTURE CAUSED BY SLOW SHRINKING

We analyze the formation of cracks induced by desiccation in terms of the following four

processes.

1. The water in a mixture evaporates from the surface of a layer.

2. Each part of the mixture shrinks upon desiccation.

3. Stress increases in the material because contraction is hindered near the bottom of a

container.

4. Fracture arises under some fragmentation condition.

In this section, we examine each process individually and construct a one-dimensional

model, where we introduce simple assumptions regarding the unclear properties of granu-

lar materials. Some similar models have been proposed previously [8,7]. One-dimensional

models assume that cracks are formed one at a time, each propagating along a line and

thereby dividing the system into two pieces separated by a boundary with one-dimensional

structure. Using this assumption, we can ignore the propagation of cracks and consider the

development of patterns by using only the condition of separation.

1. From a microscopic viewpoint, water either exists in the inside of the particles of gran-

ular materials or acts to create bonds between the particles. Here, we can introduce

the water content averaged over a much larger area than that of a single particle and

measure the degree of drying. When the thickness of a layer H is sufficiently thin and

4

the characteristic time of desiccation Td is very large, the water content in the layer

can be considered uniform. Assuming that water transfers diffusively in a layer, the

sufficient condition here is that H2/Td is much smaller than the diffusion constant.

Therefore, we restrict our consideration to the case of the uniform water distribution

and exclude the process of water transfer from the model.

2. The main cause of contraction is the shrinking of particles in the mixture arising

from desiccation. The water content is considered to determine the shrinking rate

in the case of uniform contraction in which all the boundaries of the mixture are

stress-free. We refer to this shrinking rate as “free shrinking rate” in the following

discussions and this concept is used in place of the concept of the water This makes

clear the relation between the present problems and those involving fractures induced

by other causes, such as temperature gradient [11,12], with slow contraction. We note,

however, that it is more difficult to measure the free shrinking rate than the water

content experimentally and it is thus necessary to know their relation to compare our

theory with experiments on the time development of patterns.

The contraction force is thought to arise from the water bonds among particles. We

estimate the Reynolds number Re to consider the behavior of the water in a bond. The

diameter of a particle R is generally about 0.1mm, and the kinematic viscosity of water

ν is about 1mm2/s. Although the propagation of a crack causes the displacement of

surrounding particles with opening the crack surfaces, the velocity of the displacement

is smaller than the crack speed itself, except in the microscopic region at the crack tip.

Therefore we estimate the typical velocity of water V in the bulk of a mixture to be

smaller than the crack speed. The crack speed has been measured as about 0.1 mm/s

in experiments and it is, of course, considerably faster than the shrinking speed of

the horizontal boundary with desiccation, which is typically about 10mm/day. Thus

the Reynolds number Re = RV/ν is estimated to be smaller than 1/100. We expect

that the water among the particles behaves like a viscous fluid and that the mixture

5

displays strong dissipation.

If a material displays strong dissipation and shrinks quasi-statically, the elastic field

is balanced to minimize the free energy, except during the time when cracks are prop-

agating. We deal with balanced states for the present and return to the problem of

relaxation in order to treat the development of cracks in Sec. IV. Because mixtures of

granules and fluids have many unclear properties with respect to elasticity, we idealize

them as linear elastic materials. When the free volume-shrinking rate CVis uniform,

it is well known that the free energy density of a uniform and isotropic linear elastic

material is given in terms of the stress tensor uij in the form [30],

eV=

1

2κ

V(ull + C

V)2 + µ

(

uik −1

3ullδik

)2

, (2.1)

where κVand µ are the elastic constants and repeated indices indicate summation.

The stress tensor is expressed as

σV ij ≡

∂eV

∂uij

= (λull + κVC

V)δij + 2µuij, (2.2)

where κV≡ λ+2µ/3. As a result, the balanced equation of the elastic field ∂σ

V ij/∂xj =

0 does not include the shrinking rate CV

for linear elastic materials with uniform

contraction and it is the same as in the case of the elastic materials without shrinking.

The effect of contraction appears only through the boundary conditions.

3. After the formation of cracks divides the system into cells, each cell is independent

of the others, because the vertical surfaces of cracks become stress-free boundaries.

Without considering the boundary conditions on the lateral sides of the container,

we can simplify the problem by starting with the initial condition that the system

has stress-free boundaries on its lateral sides. In contrast to the lateral and the top

surfaces, the bottom of a layer is not a stress-free boundary. The difference among

the boundary conditions produces strain with contraction and then stress. This is the

cause of fracture.

6

We observe the slip of layers along the bottom in most experiments. Thus we introduce

slip displacement with a frictional force into the model. Because the frictional force is

caused by the water between the bottom of a layer and the container, it is considered

to remain finite even in the limit of vanishing thickness of a layer H [2]. In order to

understand the effects of friction for crack patterns, we simplify the maximum frictional

force per unit area of the surface to be constant and independent of H without making

a distinction between static and kinetic frictional effects.

4. Cracks propagate very slowly in a mixture of granules and water. Because this propa-

gation resembles quasi-static growth of cracks, the first candidate of the fragmentation

condition is the Griffith criterion applied to the free energy of the entire system.

However, we note that ordinary brittle materials break instantaneously, not quasi-

statically, in the situation that the stress increases without fixing the deformation

of the system. The situation is similar to that in a shrinking mixture. We need to

consider the possibility that cracks in a mixture propagate slowly owing to dissipation.

Therefore we consider two typical fragmentation conditions, the critical stress condition

and the Griffith criterion, in the first and the last halves of Sec.III, respectively.

In the context of the critical stress condition, the fragmentation condition is that

the maximum principal stress exceeds a material constant at breaking. This has also

been used in many numerical models because of the technical advantage of the local

condition. Our model introduces the critical value for the energy density as an equiv-

alent condition. We note that the energy of the system before the fragmentation is

higher than after the fragmentation because the critical value is a material constant

independent of the system size.

In contrast, using the Griffith criterion as an alternative fragmentation condition stip-

ulates that the energy changes neither before nor after breaking. This condition is

used by Sasa and Komatsu in their theory [7].

7

With the above considerations, we construct a one-dimensional model, following the lead

of Sasa and Komatsu [7]. As we show in Fig. 1, we consider a chain of springs by a distance

a as the discrete model of the thin layer of an elastic material, where we number the nodes

i = −N,−N + 1, ..., N − 1, N for a system of half-size L := aN . In order to represent the

vertical direction of a sufficiently thin layer, we introduce vertical springs with length H

which connect each node to an element on the bottom. The vector (ui, vi) represents the

horizontal and vertical displacements of the ith node, and wi is the horizontal displacement

of the element on the bottom connected with the node. The shrinking of a material is

modeled by decreasing the natural length of the springs. We assume an isotropic material

with a linear free shrinking rate s, making the natural lengths of both the horizontal and

vertical springs to be 1− s times the initial lengths, i.e. (1− s)a and (1− s)H .

If the vertical springs are simple ordinary springs, linear response is lost under shearing

strain. We therefore add non-simple springs to the vertical direction which produce a hor-

izontal force in the case that ui 6= wi to represent a linear elastic material. This type of

spring is used in the model of Hornig et al. [8]. The energy of the system is described by

E =1

2

N−1∑

i=−N

K1(ui+1 − ui + sa)2

+1

2

N∑

i=−N

[K2(ui − wi)2 +K ′

2(vi + sH)2], (2.3)

where K1, K2 and K ′2 are the spring constants. Because vi is included in the last term

independently of both ui and wi, it follows that vi = −sH in the balanced states, and then

this term vanishes. This indicates that the model neglects the horizontal stress arising from

vertical contraction. Thus all we need to do is minimize the energy (2.3) without the last

term in order to find the horizontal displacements ui and wi. In the continuous limit, a → 0,

the above energy should be described using an independent energy density for H , as in the

case of (2.1) for a linear elastic material. We scale the space length by the thickness of

a layer H and introduce the space coordinate x := ai/H . Through the transformation to

non-dimensional variables L → LH , ui → Hu(x), wi → Hw(x) and E → H2E, the energy

(2.3) becomes

8

E =∫ L

−Ldx{e1(x) + e2(x)}, (2.4a)

e1(x) =1

2k1(ux + s)2 (2.4b)

and e2(x) =1

2k2(u− w)2, (2.4c)

and we know that both k1 :=aHK1 and k2 :=

HaK2 are the independent constants of H .

We introduce the maximum frictional force Fs for the slip of the elements on the bottom,

as explained above. The vertical spring pulls the ith element along the bottom with the

force Fi = K2(ui −wi). Each element on the bottom remains stationary if |Fi| < Fs, and, if

not, it slips to a position at which |Fi| = Fs is satisfied. The slip condition is expressed by

the energy density of a vertical spring through the following rule in the previous continuous

a → 0 limit:

e2(x) >1

2k1s

2s ⇒ w(x) = u(x)± ss

q. (2.5)

Here the choice of the sign depends on the direction of the force. The constants ss and q

are defined by the equations

1

2k1s

2s ≡

F 2s

2k2a2and q ≡

√

k2k1

. (2.6)

We note the constant q is order 1, because its square represents a ratio of certain elastic

constants which are the same order in ordinary materials.

Neglecting the short periods during which the system experiences cracking and slip, (2.4)

and (2.5) constitute the closed form of our one-dimensional model with the fragmentation

condition given in the next section.

III. ANALYSIS OF THE ONE-DIMENSIONAL MODEL

We here report analytical results of our one-dimensional model for the typical two frag-

mentation conditions, i.e, the critical stress condition and the Griffith condition.

9

A. The Critical Stress Condition

The critical stress condition demands that the maximum principal stress exceeds a ma-

terial constant at the fragmentation.

In the case of this condition, we can generally demonstrate that it is difficult to treat

the bottom surface as a fixed boundary for a uniform and isotropic elastic material. We

first explain it before the analysis of the one-dimensional model. Let us think of the layer

of a linear elastic material contracting with a fixed boundary condition on the bottom. It is

shrinking more near the top surface, and the cross section assumes the form of a trapezoid

as we show schematically in Fig. 2. We compare the stress at the following three points:

(A) the horizontal center of the cell near the bottom; (B) the lateral point near the bottom;

and (C) the horizontal center above the bottom. The horizontal tensions at A and B are the

same because of the fixed boundary condition on the bottom. Although the stress at C is as

horizontal as at A, the strength is weaker. B is also pulled in the direction along the lateral

surface due to the deformation. Using A, B, and C to represent the respective strengths

of the maximum principle stresses at these three points, we find that they are related as

C < A < B, and we expect generally that fracture arises at B before either A or C. If the

contraction proceeds while the fixed boundary condition on the bottom is maintained, the

lateral side breaks near the bottom before the division of the cell, and the fixed boundary

condition can not persist. Hence we need to consider the displacement of the layer with

respect to the bottom to deal with this problem correctly.

In our one-dimensional model, we break a spring when its energy exceeds a critical value.

We assume that the corresponding critical energy density is independent of both L and H .

As mentioned above, this is equivalent to the critical stress condition in one-dimensional

models. Representing the critical energy density with the corresponding linear shrinking

rate sb by k1s2b/2, the fragmentation conditions are described by the rules

e1(x) ≥1

2k1s

2b ⇒

The horizontal spring is cut off; the

cell is divided.(3.1)

10

e2(x) ≥1

2k1s

2b ⇒

The vertical spring is cut off; the

bottom of the layer breaks.(3.2)

Here we apply the same condition to the vertical springs in order to enforce that the lateral

side breaks before the division of a cell under the fixed boundary condition.

We can easily determine the analytical solutions. The functional variation of the energy

(2.4) on u(x) is obtained in the form

δE =∫ L

−Ldx{−k1uxx + k2(u− w)}δu

+ [k1(ux + s)δu]L−L , (3.3)

and we obtain both the balanced equation

uxx = q2(u− w) (3.4a)

and the stress-free boundary condition

ux + s = 0 at x = ±L. (3.4b)

First we assume the fixed boundary condition without slip on the bottom: w(x) = 0 for

|x| ≤ L. The solution of the equations (3.4) is then

u(x) = −s

q

sinh qx

cosh qL. (3.5)

The deformation almost only appears near the lateral boundaries, because of exponential

dumping. The energy densities of horizontal e1(x) and vertical springs e2(x) are maxima at

the center of a cell x = 0 and at the lateral boundary x = L, respectively, and these points

have the greatest possibility of breaking. Then energy densities are calculated as

e1(0) =1

2k1s

2

(

1− 1

cosh qL

)2

(3.6a)

and

e2(L) =1

2k1s

2 (tanh qL)2 . (3.6b)

11

Although they both increase with shrinking, e1(0) is always less than e2(L).

If sb is smaller than ss, that is, if breaking occurs before slip, the breaking condition

(3.2) for the vertical spring on the lateral side is the first to be satisfied. To identify the

effect of slip, we consider the fragmentation of a cell with the assumption that neither the

slip nor the breaking of the vertical springs occurs even with the fixed boundary condition.

For the first breaking of the horizontal springs, we apply the fragmentation condition (3.1)

to (3.6a) and obtain the relation between the system size and the shrinking rate:

qL = arccoshs

s− sb. (3.7)

As indicated with the solid line in Fig. 3, qL drops rapidly at s/sb ≃ 1 and then vanishes

slowly as s increases further. Because each breaking divides the system into rough halves,

the size of a cell decreases with shrinking. The figure displays the typical development of

the size of a cell with the stair-like function of the dot-dashed line and the arrows. Because

the system size L is scaled by the thickness H , we see that the system is divided into a size

smaller than H after sufficient shrinking.

If sb is larger than ss, the layer starts to slip from the lateral sides when e2(L) = k1s2s/2.

The shrinking rate at that time is given by

s =ss

tanh qL. (3.8)

We next investigate this case.

We suppose that the symmetrical slip from both lateral sides is directed toward the

center and only consider the half region x > 0. The function w(x) becomes finite in the slip

region xs < x ≤ L and remains zero elsewhere, where we introduce xs as the starting point

of the slip region. The slip displacement w(x) ≡ w0(x) is expressed by the displacement

u(x) ≡ u0(x) as

w0(x) =

0 0 < x ≤ xs

u0(x) +ssq

xs < x ≤ L. (3.9)

12

Equation (3.4) then take the form

u0xx =

q2u0(x) 0 < x ≤ xs

−qss xs < x ≤ L, (3.10a)

b.c.: u0x + s = 0 at x = L , (3.10b)

and the matching conditions are

w0(x), u0(x) and∂u0(x)∂x

are continuous at x = xs. (3.10c)

We derive the solution u0(x) in each region and obtain

u0(x) =

A sinh qx 0 ≤ x < xs

[

qss(

L− 12x)

− s]

x+B xs ≤ x < L. (3.11)

The three matching conditions give the integral constants A and B and yield the equation

to determine xs,

q(L− xs) =s

ss− 1

tanh qxs

. (3.12)

At xs = L, this reduces to (3.8). This form can be approximated as L − xs ≃ (s− ss)/qss

for qxs ≫ 1 and as qxs ≃ (s/ss − qL)−1 for qxs ≪ 1.

We calculate the energy density e1(0) again and substitute this into the breaking condi-

tion (3.1). This gives the equation

s

ss− 1

sinh qxs

≥ sbss. (3.13)

Eliminating xs from the equations (3.12) and (3.13), we obtain the following relation

between the system size and the shrinking rate at the first breaking:

qL = arcsinh(

sss− sb

)

+s

ss−√

1 +(

s− sbss

)2

. (3.14)

We see that qL is a decreasing function of s. It decreases slowly to the limiting value sb/ss

after the rapid drop in the range sb ≤ s <∼ sb + ss.

13

Figure 4 exhibits two curves of the shrinking rates at the start of slip (3.8) and at the

first breaking (3.14), where half of the system size L is represented on the vertical axis as

in Fig. 3. After full contraction, the final size of a cell is close to the asymptotic value of

the curve defined by (3.14), qL ∼ sb/ss. The region without slip also becomes smaller, and

its final size is given by qxs ≃ ss/(s − sb), where we assume qxs ≪ 1 in (3.12). With the

original scale, we obtain

L ≃ H

q

sbss

and xs ≃H

q

sss− sb

for s− sb >∼ ss. (3.15)

Thus L and xs are proportional to the thickness of a layer H , although there is the possibility

for them to be modified through ss if the frictional force depends on H .

The first equation of (3.15) is consistent with the experimental results of Groisman

and Kaplan [2] for the final size of a cell after full desiccation, as mentioned in Sec. I.

The assumptions used in this analysis are also consistent with those in their qualitative

explanation, where they considered the balance between the frictional force and elastic force

[2]. In addition, when we peel the layer of an actual mixture after drying, we often observe

a circular mark at the center of each crack cell on the bottom of the container. Its size is

approximately equal to the thickness of the layer. We can understand these marks as the

sticky region |x| < xs.

B. The Griffith Criterion

Next we apply the Griffith criterion [29] to the entire system as the fragmentation con-

dition in the place of the critical stress condition. This was used by Sasa and Komatsu in a

different model [7].

First we again assume the fixed boundary condition, where neither the slip nor the

breaking of the vertical springs occurs. The Griffith criterion introduces the creation energy

of a crack surface per unit area Γ and assumes the cracking condition that the sum of the

creation energy and the elastic energy decreases due to breaking. We write the elastic energy

14

of a system −L ≤ x ≤ L as E(2L). We consider the case in which the cell with size 2L (the

system size) is divided into exact halves. The alternative fragmentation condition to (3.1)

is given by the equation

∆E(2L) ≡ E(2L)− 2E(L) ≥ ΓH. (3.16)

We calculate (2.4) by using (3.5) to obtain the elastic energy E(2L) for the fixed boundary

condition. With the original scale, it is given by

E(L) = k1s2HL

(

1− H

qLtanh

qL

H

)

. (3.17)

As a result, we obtain the following relation in the place of (3.14) for the shrinking rate

at the first breaking:

∆E(L) =(

sΓs

)2

and sΓ ≡√

qΓ

k1H. (3.18)

Here,

∆E(L) ≡ q∆E(L)

k1s2H2= 2 tanh

qL

2H− tanh

qL

H

=

1 qL ≫ H

14

(

qLH

)3qL ≪ H

. (3.19)

The corresponding curve is indicated with the dotted line in Fig. 3, where the shrinking rate

s is scaled by sΓ. This curve agrees quite well with the solid line representing the previous

results (3.7), so we again find that cells are divided into a size smaller than H after breaking.

We however note that sΓ depends on the thickness H , although both sb and sΓ represent the

shrinking rate at the first breaking for an infinite system. Because the ratio of the surface

energy Γ to the elastic constant is a microscopic length for ordinary materials, sΓ is inferred

to be very small. Therefore, with the Griffith criterion, we usually expect that sΓ is smaller

than ss and no slip occurs before breaking.

Next we show that, even if sΓ is larger than ss, the Griffith criterion does not yield the

proportionality relation of the final size of a cell to the thickness of the layer H . Elastic

15

energy is consumed not only by the creation of the crack surface but also by the friction due

to slip on the bottom. We again consider the breaking of the system (−L < x < L) into

exact halves. An alternative Griffith condition is given by

∆Es(L) ≡ Es(2L)− 2[E ′s(L) +Ws] ≥ ΓH, (3.20)

where Es(2L) and 2E ′s(L) represent the elastic energies of the system before and after

breaking, respectively, and 2Ws is the work performed by the frictional force due to slip.

As the state just before breaking, we consider a cell with symmetric slip regions. This

state has been derived in (3.9), (3.11) and (3.12). The elastic energy Es(2L) is obtained by

calculating (2.4) in the form

Es(2L) =k1s

2s

q

[

1

3q3(L− xs)

3 +(

s

ss

)2

qxs −s

ss

]

, (3.21)

where the width of the slip region L−xs is determined as a function of L and s/ss by (3.12).

In order to estimate Ws and E ′s(L), we need to investigate the detailed process of frag-

mentation. Here we imitate an actual quasi-static fracture in two dimensions by using a

hypothetical quasi-static process in the one-dimensional model. We introduce a traction

force on crack surfaces which prevents the crack from opening and obtain the final state of

this process with the stress-free boundaries by stipulating that the strength vanishes quasi-

statically. The work of the hypothetical traction force is considered to be the opposite of

the creation energy of the crack. Let us imagine the right half 0 < x < L just after the

breaking at the center x = 0, where the traction force works at x = 0 to the left. Because

of the relaxation of the traction, the slip region xs < x < L before the breaking vanishes

immediately. As the traction decreases, a new slip region is created in 0 < x < xr on the

side of the crack. If the contraction ratio s is much larger than ss, we may assume that xs is

smaller than xr at the end of this process, because qxs<∼ 1, and the new slip region expands

to xr ≃ L/2. The slip displacement w(x) in the state is given by the initial condition (3.9)

and the slip condition (2.5) as

16

w(x) =

w0(x) xr < x < L

u(x)− ssq

0 < x < xr

. (3.22)

The solution of (3.4) is

u(x) =

u0(x) + C ′1 cosh q(x− L) xr < x < L

12qssx

2 − sx+ C ′2 0 < x < xr

, (3.23)

where the conditions of the continuity of w(x), u(x) and ux(x) at x = xr determine the

constants C ′1 and C ′

2 and produce the equation for xs:

q(L− 2xr) = 2 tanh q(L− xr). (3.24)

We calculate the elastic energy (2.4) from (3.22) and (3.23) and obtain the energy at the

end of this process,

2E ′s(L) =

k1s2s

q

{

1

3q3[x3

r + (L− xr)3]− qL

}

. (3.25)

Because slip occurs with a constant frictional force from the previous assumption, the work

Ws can be expressed by the integral of the total distance of slip,

Ws =Fs

a

∫ L

0dx|w(x)− w0(x)|, (3.26)

and it is calculated from (2.6), (3.22) and (3.23) as

2Ws =k1s

2s

q

[

2

3q3(x3

s − 2x3r) + q3Lx2

r −(

qL− s

ss

)

q2x2s

]

. (3.27)

In order to know the scaling relation of the final size of a crack cell after full desiccation,

we assume qL ≫ 1 and the limit of the full contraction: s/ss → ∞. Because (3.12)

and (3.24) give the approximate equations qxs ≃ ss/s ≪ 1 and xr ≃ L/2, respectively,

(3.20),(3.21),(3.25) and (3.27) result in the equation

∆Es(L) ≃k1s

2s

6q(qL)3. (3.28)

With the original scaling, the Griffith criterion (3.20) gives the scaling relation of the final

size of a cell for the thickness H ,

17

qL >∼3√6H

(

sΓss

) 2

3 ∝ H2

3 , (3.29)

where we use sΓ defined in (3.18), and the condition sΓ ≫ ss is necessary from the assumption

qL ≫ 1. Thus the Griffith criterion gives the different scaling relation because of the

dependence of sΓ on H , although we obtained the proportionality relation (3.15) under the

critical stress condition.

As a result, the critical stress condition and the Griffith condition lead to different

relations between the final size of a crack cell and the thickness of a layer. Experimental

results seem to support the former. Although the results should be discussed further, of

course, they suggest the possibility that the dissipation in the bulk can not be neglected for

the fracture of a mixture of granules and water.

IV. THE DEVELOPMENT OF CRACKS IN TWO DIMENSIONS

The one-dimensional model we have discussed to this point idealizes the process of crack-

ing, treating cracks as one-dimensional structures forming one at a time parallel to one an-

other. In order to consider the development of cracks and their pattern formation, we must

extend this model to two dimensions and include the relaxation process of the elastic field.

Although mixtures containing granular materials that are rich with water generally pos-

sess viscoelasticity, visible fluidity can not be observed at the time of cracking after the

evaporation of water with desiccation. Therefore, we assume that only the relaxation of

strain contributes to the dissipation process in a linear elastic material. For simplicity, we

assume that the system has the only one characteristic relaxation time.

In the one-dimensional model, the total energy (2.4) consists of terms representing a

one-dimensional linear elastic material and the vertical shearing strain, the both of which

are quadratic in the displacements. Expanding the elastic material to the horizontal xy

18

plain, we naturally obtain the extended energy in two dimensions as

E =∫

dxdy{e1(x, y) + e2(x, y)}, (4.1a)

e1(x, y) ≡1

2κ(ull + C)2 + µ

(

uik −1

2ullδik

)2

(4.1b)

and e2(x, y) ≡1

2k2(u−w)2, (4.1c)

where uij ≡ (ui,j + uj,i)/2 and ui,j ≡ ∂ui/∂xj . In analogy to the one-dimensional model,

the two-dimensional vector fields u(x, y) and w(x, y) represent the displacement of a layer

from the initial position and the slip displacement on a bottom, respectively. The space

coordinates x and y and the displacements u(x, y) and w(x, y) are again scaled by the

thickness of a layer, H . The expression in (4.1b) is the energy density of the two-dimensional

linear elastic material with a uniform free surface-shrinking rate C.

The shrinking speed C can be neglected from the assumption of quasi-static contraction.

The time derivative of (4.1) is given by

E =∫

dxdy{[−σij,j + k2(ui − wi)]ui

−k2(ui − wi)wi}+∮

dSnjσij ui, (4.2a)

σij ≡ (λull + κC)δij + 2µuij and κ ≡ λ+ µ, (4.2b)

where∮

dS represents the integral along the boundary of the cell and n is its normal vector.

The dissipation of energy arises from the non-vanishing relative velocities of neighboring

elements in a material, and then the time derivative E can also be represented as a function

of them. As is well known, E can be written in a form similar to the energy due to certain

symmetries [30]. We have

E = −∫

dxdy{κ′u2ll + 2µ′

(

uik −1

2ullδik

)2

+k′2(u− w)2}

= −∫

dxdy{[−σ′ij,j + k′

2(ui − wi)]ui

−k′2(ui − wi)wi} −

∮

dSnjσ′ij ui (4.3a)

19

to the second order, where

σ′ij ≡ λ′ullδij + 2µ′uij and κ′ ≡ λ′ + µ′. (4.3b)

Although the constants κ′, µ′ and k′2 are generally independent of κ, µ and k2, we assume

they take simple forms with one relaxation time τ , writing σ′ij = τ∂σij/∂t, or equivalently,

κ′ = τκ, µ′ = τµ and k′2 = τk2. (4.4)

Equations (4.2), (4.3) and (4.4) yield the time evolution equation of u(x, y),

(

1 + τ∂

∂t

)

[σij,j − k2(ui − wi)] = 0, (4.5)

and the free boundary condition,

(

1 + τ∂

∂t

)

σijnj = 0. (4.6)

As mentioned above, the shrinking rate C only appears in the free boundary condition.

Except for the effect of shrinking and slip, the above equations are essentially the same

as the Kelvin model, proposed for viscoelastic solids. Because the existence of a bottom

causes a screening effect through the term k2ui, the elastic field decays exponentially in the

range of the thickness of a layer, i.e, the unit length in (4.5). Here the stress in the material

is (1 + τ∂/∂t)σij by adding the dissipative force. With the definitions

Ui ≡(

1 + τ∂

∂t

)

ui, Wi ≡(

1 + τ∂

∂t

)

wi (4.7a)

and

Σij ≡(

1 + τ∂

∂t

)

σij , (4.7b)

(4.5) and (4.6) can be rewritten as

Σij,j = k2(Ui −Wi), (4.8a)

Σij = (λUll + κC)δij + 2µUij, (4.8b)

20

doing with the free boundary condition

njΣij = 0. (4.8c)

Therefore, Ui satisfies the balanced equations of an ordinary elastic material without dissi-

pation.

We expect that the state of the water bonds in a mixture can be represented by σij ,

i.e. the stress excluding the dissipative force, rather than by the stress Σij itself because

σij is a function of the strain uij. For this reason we introduce the breaking condition by

using σij in the following analysis. The propagation of a crack in the Kelvin model has been

investigated by many peoples [21–25,28]. Although the stress field diverges at a crack tip

in the continuous Kelvin model, it is possible that the divergence of σij is suppressed by

the advance of a crack. We calculate the elastic field around a crack tip for a straight crack

which propagates stationarily in an infinite system. Because near the tip of a propagating

crack there is little slip, as the simulations in the next section indicate, we can assume

w ≃ 0, and then Wi(x, y) = 0 in (4.8). Although our model is incomplete in the sense

that the divergence of the stress can not be removed in continuous models of a linear elastic

material, we expect that the following discussions are valid.

We consider a straight crack with velocity v that coincides with the semi-infinite part of

the x-axis satisfying x < vt in two-dimensional plane. The stress field satisfies the stress-free

boundary conditions on the crack surface, and the displacement u vanishes as |y| → ∞. We

define the moving coordinates (ξ, y) as ξ ≡ x − vt and assume reflection symmetry on the

x-axis. The boundary conditions are given by

Σyy = 0 on y = 0, ξ < 0

Uy = 0 on y = 0, ξ > 0

Σξy = 0 on y = 0

Ui → 0 for |y| → ∞

. (4.9)

21

The stress field under the above boundary conditions can be obtained by the Wiener-Hopf

method. Fortunately, this problem reduces to the following solved problem for the mode

I type of a crack. We consider a stationary crack along the negative x-axis in completely

linear elastic material without contraction, where we include the inertial term with the mass

density ρ. When a uniform pressure σ∗ is added on the surface of the crack from time t = 0,

the stress field u0i is given by the equations

σ0ij,j = ρ

∂2u0i

∂t2and σ0

ij ≡ λu0llδij + 2µu0

ij (4.10)

and the boundary conditions

σ0yy = −σ∗Θ(t) on y = 0, x < 0

u0y = 0 on y = 0, x > 0

σ0xy = 0 on y = 0

u0i → 0 for |y| → ∞

. (4.11)

These become identical to the previous equations when we apply the Laplace transformations

on time,

Ui(x, y, η) =∫ ∞

0dtu0

i (x, y, t)e−ηt (4.12a)

and

Σ0ij(x, y, η) =

∫ ∞

0dtσ0

ij(x, y, t)fe−ηt, (4.12b)

and make the replacements η2ρ = k2, σ∗ = κCη, Σij = Σ0

ij + κCδij and x → ξ. Here η can

be taken equal to 1 because the correspondence holds for any η.

Using the analytical solutions [32] of (4.10) and (4.11), we shall consider the stress in

front of the crack, i.e. Σyy(ξ, 0) where ξ > 0. The above replacements give the solution of

our problem as,

Σyy(ξ, 0) = κC{

−1

π

∫ ∞−0i

α−0idζ ImΣ0(−ζ) e−ζξ + 1

}

, (4.13)

α ≡√

k2λ+ 2µ

and Σ0(ζ) ≡ 1

ζ

(

F+(0)

F+(ζ)− 1

)

(4.14)

22

where F+(ζ) is an analytical function in the region Reζ > −α such that

F+(ζ) → ζ−1

2 for |ζ | → ∞ (4.15)

and

F+(0) =

√

√

√

√

2µ(λ+ µ)

α(λ+ 2µ)2. (4.16)

Because ImΣ0(−ζ) approaches −F+(0)/√ζ as |ζ | increases, it can be approximated in the

region αξ ≪ 1 around the crack tip by the equation

Σyy(ξ, 0) ≃ κC

(

F+(0)

π

∫ ∞

0dζζ−

1

2 e−ζξ + 1

)

= κC

(

F+(0)√πξ

+ 1

)

. (4.17)

We see that the stress Σyy diverges in inverse proportion to the square root of the distance

near the tip.

Solving (4.7b) in the moving system ξ = x− vt, we get

σyy(ξ, 0) =1

τv

∫ ∞

ξdξ′Σyy(ξ

′, 0)eξ−ξ′

τv , for ξ > 0. (4.18)

and the following equation is obtained by substituting (4.13) into this equation:

σyy(ξ, 0) = κC

{

−1

π

∫ ∞−0i

α−0idζ

ImΣ0(−ζ)

τvζ + 1e−ζξ + 1

}

. (4.19)

We first examine the behavior of σyy in the region αξ ≫ 1 far away from the tip. Because

the integral in (4.19) can be approximated as

∫ ∞−0i

α−0idζ

ImΣ0(−ζ)

τvζ + 1e−ζξ

=

[

∫ ∞

0dζ

ImΣ0(−ζ − α)

τv(ζ + α) + 1e−ζξ

]

e−αξ

≃ ImΣ0(−α)

τvα + 1

e−αξ

ξ, (4.20)

σyy decays exponentially to the uniform tension as

σyy(ξ, 0) ≃ κC

(

D

ξe−αξ + 1

)

for αξ ≫ 1, (4.21a)

23

where

D ≡ − ImΣ0(−α)

π(τvα + 1). (4.21b)

The characteristic length of the decay is H/α with the original scale, which is of the same

order as the thickness H for an ordinary elastic material.

In the region αξ ≪ 1 near the tip, we approximate the integral with the asymptotic form

of ImΣ0(−ζ) for large ζ as

− 1

π

∫ ∞−0i

α−0idζ

ImΣ0(−ζ)

τvζ + 1e−ζξ

≃∫ ∞

0dζ

F+(0)

π(τvζ + 1)ζ1

2

e−ζξ

=2F+(0)√

πτve

ξ

τv erfc

√

ξ

τv

, (4.22a)

erfc(x) ≡∫ ∞

xdte−t2 ≃ 1

2xe−x2

for x → ∞, (4.22b)

and erfc(0) =

√π

2, (4.22c)

and then we obtain the equations

σyy(ξ, 0) ≃

κC(

F+(0)√πξ

+ 1)

ξ ≫ τv

κC(

F+(0)√τv

+ 1)

ξ ≪ τv

for αξ ≪ 1. (4.23)

With the original scale again, σij is almost proportional to√

H/ξ in the middle region from

τv to H/α around the crack tip, as is the case for the stress Σij . However, σij is bounded

at the tip by the proportional value to√

H/τv.

Thus the stress excluding the dissipative force, i.e. σij , is kept from diverging by the

movement of the crack tip. Therefore we can introduce a critical value σth as a material

parameter again and assume the breaking condition at the tip

limξ→0+

σyy(ξ, 0) ≥ σth ≡ κCth, (4.24)

24

where the constant Cth represents the shrinking rate corresponding to the critical value.

For a stationary propagating crack, we find the velocity v by substituting σyy(ξ, 0) into the

above equation in (4.23) as

v = F+(0)2(

C

Cth − C

)2 H

τ. (4.25)

Although this equation is not valid near the sound velocity because we have neglected inertia,

we expect that a material cracks at a shrinking rate C below Cth due to inhomogeneity. Thus

(4.25) explains why the propagation velocity observed in experiments is very small compared

to the sound velocity. In addition, it suggests the proportionality relation between the

thickness and the propagating speed, which should be experimentally observable.

Next we note the validity of (4.25) for very slow speeds. Although the divergence of σyy

is suppressed by the advance of a crack, as we see in (4.23), the size of the screening region

is approximately τv. Because particles of size R = 0.01 ∼ 1mm are used in experiments, the

above breaking condition (4.24) is available for velocities v ≫ R/τ , where the continuous

approximation is valid. For very slow velocities, τv ≪ R, for example, it may be possible

for the defects in a material to arrest the growing of cracks.

The speed of a real crack measures v <∼ 2mm/min for the thickness of a layer of coffee

powder, H ≃ 6mm, as observed in experiments [2]. Although we have not yet specified

the origin of the relaxation, we attempt to estimate the relaxation time τ arising from the

viscosity of the water in the bonds among particles. Supposing that the diameter of a particle

of the coffee powder is about R ∼ 0.5mm and the viscosity of the water is ν ∼ 1mm2/s, we

obtain τ ∼ R2/ν ∼ 0.25s and τv/R ∼ 0.02 < 1. This rough estimate suggests the possibility

that the above continuous approximation is imperfect in the case of a relatively thin layer.

Groisman and Kaplan reported the interesting experimental results that the propagation

speed exhibits a wide dispersion even among cracks growing at the same time, although the

speed of each individually is almost constant on time. It is a future problem to understand

the relation of the dispersion to the inhomogeneity of materials.

25

V. THE PATTERNS OF CRACKS

Cracks appear one after another with shrinking, and spread over the system to create

a two-dimensional pattern. In this section, we report on a study of the formation of crack

patterns using numerical simulations of the two-dimensional model introduced in the pre-

vious section. We make the natural extension of both the breaking condition and the slip

condition employed in the one-dimensional model by using the energy densities defined in

the microscopic cells of the lattice in the simulations. In the two-dimensional model, the for-

mer is simpler than the critical stress condition because it neglects the direction of both the

stress and the microscopic crack surfaces. In addition, the slip condition implicitly assumes

a sufficiently short period of slip in comparison to the relaxation time of the elastic field

because of the balance between the frictional force and elastic force. We report the results

of our simulations after describing the discrete method and these extended conditions.

As is the case with most fingering patterns, the growing of cracks is influenced strongly

by the anisotropy of the system. We used random lattices [33,34] in our simulations to

consider uniform and isotropic systems in a statistical sense.

Many fracture models employ a network of springs or elastic beams to model an elastic

material [35–38,14,15,8,10,39,40]. However, it is generally difficult to calculate the elastic

constants for an elastic material modeled by such a lattices. Therefore, instead of such

networks, we consider each triangular cell in a random lattice as a tile of the elastic material

with uniform deformation. A fracture is realized by removing any cell whose energy density

exceeds a critical value, as we explain below.

We construct a two-dimensional model as is illustrated in Fig. 5. Each site of the lattice

is connected to an element on the bottom with a vertical spring similar as that used in the

one-dimensional model. Figure 6 displays a part of the random lattice which represents a

horizontal elastic plane. The random lattice is composed of random points to form a Voronoi

division of them. We number the sites and the triangular cells in the lattice and express

26

the area of the Voronoi cell around the nth site as Vn (n = 1, 2, 3, ..., N) and the area of the

mth triangle as Tm (m = 1, 2, 3, ..., NT ).

The displacement of the nth site u(n)(t) is obtained from the U(n)(t) by the definition

(4.7a). A simple Euler method gives the following equation with discrete time ∆t:

u(n)(t+∆t) = u(n)(t) +∆t

τ(U(n)(t)− u(n)(t)). (5.1)

Because U(t) satisfies the balanced equation of an ordinary elastic material at any time,

it minimizes the energy E defined by (4.1) through the replacements u → U and w → W.

E consists of the energies of both the vertical springs and the horizontal elastic plain. We

calculate the latter by summation of the energies of the triangular cells in the random lattice,

where the mth triangular cell is assumed to consist of a linear elastic material with uniform

strain tensor U(m)ij . Thus we obtain the equations

E =NT∑′

m=1

Tme(m)1 +

N∑

n=1

Vne(n)2 , (5.2a)

e(m)1 ≡ 1

2κ(U

(m)ll + C)2 + µ

(

U(m)jk − 1

2U

(m)ll δjk

)2

(5.2b)

and e(n)2 ≡ 1

2k2(U

(n) −W(n))2, (5.2c)

where U(n) and W(n) represent the displacement of the nth site and the slip displacement

of the nth element along the bottom, respectively. Although the rule of repeated indices is

applied to j, k and l, as usual, the summations over m and n are expressed by the symbol

∑

, where∑′

m represents a summation that excludes broken triangle cells.

The quantity e(m)1 is the elastic energy of the mth triangle cell which is calculated from

U(m)ij . For the following explanation, we express the vertices of the mth triangle as n = 1, 2, 3

and their initial equilibrium positions as x(n) ≡ (x(n), y(n)), as is shown in Fig. 6. Assuming

uniform deformation in the triangle, the strain tensor U(m)jk ≡ 1

2(U

(m)j,k +U

(m)k,j ) is given by the

displacements of the vertices U(n) as the equations

U (m)xx =

1

Tǫijky

(ij)U (k)x , (5.3)

U (m)yy = − 1

Tǫijkx

(ij)U (k)y , (5.4)

27

U (m)xy =

1

2Tǫijk[y

(ij)U (k)y − x(ij)U (k)

x ], (5.5)

where

T ≡ ǫijkx(i)y(jk), Tm =

1

2|T |, x(ij) ≡ x(i) − x(j), (5.6)

and ǫijk is Eddington’s ǫ. These equations are easily obtained from the first order Taylor

expansions of U(n) = U(x(n), t) in the triangle. Thus we can calculate the energy (5.2) on

the random lattice and obtain U(n) from its minimum.

A fracture is represented by the removal of triangle cells, not by the breaking of bonds,

in this model. This gives a direct extension from the one-dimensional model, although

it neglects the microscopic direction of the stress and the cracking in a triangle cell. We

calculate the elastic energy density of themth triangular cell e(m) from the true displacements

u(n), and assume a critical value for the breaking, which is represented by the corresponding

shrinking rate Cb as κC2b /2. Thus the breaking condition is given by

e(m)1 ≥ 1

2κC2

b ⇒The mth triangle cell is re-

moved,(5.7)

where

e(m)1 ≡ 1

2κ(u

(m)ll + C)2 + µ

(

u(m)jk − 1

2u(m)ll δjk

)2

, (5.8)

and u(m)jk is calculated from u(n) using the method explained above for U

(m)jk .

The slip condition for the elements on the bottom is also similar to that in the one-

dimensional model. We introduce the maximum frictional force as a constant and assume

the balance of the fictional force against the sum of the elastic force and the dissipative

force, i.e., |k2 (1 + τ∂/∂t) (u −w)| = k2|U−W|. Therefore, the slip condition for the nth

element can be written by using e(n)2 as

e(n)2 ≥ 1

2κC2

s ⇒

W(n) is moved along the

force to the position where

e(n)2 = 1

2κC2

s .

(5.9)

28

The slip displacement w(n) is calculated from W(n) by the definition (4.7a) as

w(n)(t +∆t) = w(n)(t) +∆t

τ(W(n)(t)−w(n)(t)). (5.10)

We carried out the numerical simulations of the model by increasing the shrinking rate

C in proportion to time t with a constant rate C. We repeated the following procedures at

each time step:

1. The contraction rate increases as C(t +∆t) = C(t) + ∆tC .

2. The displacement u is calculated from U which is the function minimizing the energy

(5.2).

3. The slip w is calculated from W, which is given by the conditions (5.9) at the all sites.

4. If some triangular cell satisfy the breaking condition (5.7), it is removed. Its energy is

not included in subsequent calculations.

If more than one cell satisfies the breaking condition at step 4, we repeat the step 1-3 using

a smaller time step.

In the above calculations, we can take the parameters κ, k2, Cb and C to be 1 with loss

of generality by scaling space, time, energy and the shrinking rate as follows,

x →√

κ

k2x, u → Cb

√

κ

k2u, w → Cb

√

κ

k2w,

E → 1

2κC2

bE, t → Cb

Ct, and C = CbC. (5.11)

Here we write the scaled shrinking rate as C. As a result, three independent parameters

remain explicitly in the equations:

µ ≡ µ

κ, τ ≡ C

Cb

τ, and Cs ≡Cs

Cb

. (5.12)

Although the properties of the lattice influence the results, we used an identical lattice

in all of our simulations, except the last in which we consider the effect of a random lattice.

29

To prepare the sites in the random lattice, we arranged points in a triangular lattice with

mesh size 0.01 inside a square region 1 × 1 and shifted the x, y coordinates of each point

by adding uniform random numbers within the range ±0.005. Because their distribution is

almost uniform and random, we connect them by Voronoi division to make the network of

the random lattice. Then the square region is extended to the size L × L, which is related

to the original system size L as

L ≡√

k2κ

L

H. (5.13)

We use the conjugate gradient method [41] with a tolerance 10−6 to find the minimum

points of functions on free boundary conditions. The time step ∆t is changed automatically

in the range to an upper bound 10−4−10−3. In our simulations, at most one cell is removed

in any given timestep. In contrast, we note that in the model without a relaxation process,

a crack propagates instantaneously at a fixed shrinking rate, because the breaking of a cell

necessarily changes the balanced state of the elastic field and often causes a chain consisting

of the breaking of many cells occuring simultaneously.

We show the typical development of a crack pattern in our model in Fig. 7, where the

parameters are L = 10√2, µ = 1, ˆτ = 0.01 and Cs = 0.5, and the black area indicates

the removed triangular cells. The gray scale represents the energy densities (5.8) in Figs.

7 (a),(b) and (d), and each dot in Fig. 7 (c) is a slipping element under the condition

(5.9). C = 1 (i.e. C = Cb) corresponds to the shrinking rate at the first breaking for an

infinite system. The first breaking in the simulations occurs slightly below C = 1 for most

parameters because of the randomness in the lattice. At that time, no slip has yet begun in

the most of the system, except near the boundaries.

After this, some crack tips grow simultaneously in the whole system, and the crack

pattern almost becomes complete at the shrinking rate near C = 1. Here we see the white

circular marks around the center of the crack cells, as shown in Fig. 7-(c). These represent

the sticky regions without slip. Similar marks can be observed on the bottom of a container

30

in actual experiments, as we mentioned in Sec. III.

Figure 8 graphs the development of the total energy (4.1) with shrinking for the three

cases τ = 0.1, 0.01 and 0.001. The energy increases with contraction. However, it is released

and dissipates due to successive breaking and becomes almost constant with increasing

shrinking rate. Our simulations were carried out until C = 10. The crack patterns changed

little when C becomes large, while the circular marks shrank gradually.

For fast relaxation, new cracks grow from the lateral side of another crack almost per-

pendicularly. Figure 9 displays a snapshot of a crack pattern for the very small relaxation

time τ = 0.001. As τ becomes smaller, the cracks tend to propagate faster and grow one

at a time, in agreement with the assumption in the one-dimensional model. In Fig. 10,

we compare how the total number of broken triangular cells increases with shrinking for

τ = 0.1, 0.01 and 0.001. The number of broken cells is almost proportional to the total

length of cracks. It increases like a step function for τ = 0.001. This indicates that the

cracks are formed one by one.

For slow relaxation, in contrast, the growing cracks from fingering-type patterns [35–38],

such as similar to those seem in viscous fingering. As is shown in Fig. 11, many cracks tend

to grow simultaneously from the center toward the boundaries. They are accompanied by a

series of tip splittings and the total length of the cracks increases smoothly with contraction.

As τ becomes larger, it takes more time to complete the crack patterns because of the slower

propagation of cracks.

We can see the influence of slip on the crack patterns by changing Cs with the other

parameters fixed. Figures 12 and 13 are snapshots of a crack pattern after full shrinking:

C = 10 for Cs = 1.0 and Cs = 0.1, respectively. Figure 14 shows the total number of broken

triangular cells for the various values of Cs. This figure indicates that the crack patterns at

C = 10 are close to final states. As we expect, the final size of a cell becomes larger with

smaller Cs. Figure 15 plots the total numbers of broken triangles at C = 10 for Cs. They

31

increase monotonously in this range with an almost constant rate.

Next we change the elastic property with µ. From the equations of σij (4.23) and the

crack speed (4.25) in Sec. IV, we expect that, as µ becomes smaller, the stress, excluding

the dissipative force, has a weaker concentration at a crack tip, and the cracking speed is

smaller. Figure 16 shows a snapshot of a crack pattern for µ = 0.02. We find that the cracks

become irregular and jagged lines and that they propagate slowly. The crack patterns also

reach completion more and more slowly as µ decreases, as is shown in Fig. 17.

All of the above simulations were executed on the same random lattice. For comparison,

we also performed a simulation using a regular triangular lattice in the place of the random

lattice. Figure 18 shows the result using the same parameters used in Fig. 7 except for the

difference of the lattices. It is clear that the anisotropy of the triangular lattice is reflected

in the direction of cracks.

Thus we can reproduce patterns similar to those of actual cracks using our two-

dimensional model. The dependence of the formation of cracks on the relaxation time

and the elastic constants should be compared with actual experiments in more detail. The

experimental results of Groisman and Kaplan suggest a transition in the qualitative nature

of patterns as the thickness is changed. This is point (3) mentioned in Sec. I. Similar

changes of patterns are observed for slow relaxation or small µ in our simulations. However,

our model does not contain the thickness H explicitly because of the scaling with H , and we

have no experimental data for the dependence of the other parameters on H . In addition,

it is possible that the inhomogeneity in a material plays an important role in this change

because the size of a particle can become significantly large if H is made sufficiently small

[5]. Obtaining a more detailed understanding that address these points is left as a future

project.

32

VI. CONCLUSIONS

We studied the pattern formation of cracks induced by slow desiccation in a thin layer.

Assuming quasi-static and uniform contraction in the layer, we constructed a simple model

in Sec. II. It models the layer as a linear elastic plane connected to elements on the bottom

and considers the slip with a constant frictional force.

In Sec. III, we considered the critical stress condition by introducing a critical value of

the energy density. This model explains the proportionality relation between the final size

of a crack cell and the thickness of a layer and the experimental observations on the effect

of slip. We also considered the Griffith criterion as an alternative fragmentation condition.

However it predicts qualitatively different results for the final size of a crack. Although these

results need more detailed considerations, they suggest the possibility that dissipation in the

bulk may be important in these materials.

In Sec. IV, we extended the model to two dimensions and introduced the relaxation

of the elastic field to describe the development of cracks. This is essentially the same as

the Kelvin model for viscoelastic materials. Because the stress excluding the dissipative

force does not diverge at the tip of a propagating crack, we introduced a critical value as

the breaking condition in front of a moving crack. Assuming the existence of a stationary

propagating crack, we obtained an estimation for the crack speed in closed form within a

continuous theory. This estimation explains the very slow propagation of actual cracks and

predicts the proportionality relation between the crack speed and the thickness of a layer.

It is an open problem to understand the origin of the dissipation we introduced intuitively

and the role of inhomogeneity in the stability of a crack.

In Sec. V, we carried out numerical simulations of the two-dimensional model to investi-

gate the formation of crack patterns. By using the energy density defined on the microscopic

cells of a lattice, we introduced a simplified breaking condition from the direct extension of

the one-dimensional model. We used a random lattice to remove the anisotropy of the lattice

and obtained patterns similar to those observed in experiments. We found that cracks grow

33

in qualitative different ways depending on the ratio of the elastic constants and the relaxation

time. It is important in the connection with fingering patterns that for the slow relaxation

crack patterns are formed by a succession of tip-splittings rather than by side-branching.

We need a better understanding of the role of inhomogeneity to explain the transition in

the nature of patterns as a function of the thickness reported in the experiments.

For the experimental results of Groisman and Kaplan, which we mentioned in the points

(1)-(3) in Sec. I, we believe the present results give qualitative explanations for (1) and (2)

and a clue for (3), although more considerations is necessary.

ACKNOWLEDGEMENTS

The paper of S. Sasa and T. Komatsu first motivated the author and is the starting point

of this study. The author had fruitful discussions with T. Mizuguchi, A. Nishimoto and Y.

Yamazaki, who are also coworkers on project involving an experimental study of fracture.

The critical comments of S. Sasa and H. Nakanishi led to a reevaluation of the study. G.C.

Paquette are acknowledged for a conscientious reading of the manuscript. Lastly the author

is grateful to T. Uezu, S. Tasaki and the other members of our research group.

34

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37

FIGURES

H

a

ui vi

wi

( ),

ii-1

1

2

x

z

FIG. 1. The one-dimensional model

A

B

C

FIG. 2. The cross section of a shrink-

ing cell of an elastic material.

0 1 20

10

20critical stress cond.Griffith criterion

s/sΓ

qLH

crack

s/sb or

FIG. 3. The shrinking rate at the first

breaking under the fixed boundary condi-

tion is plotted as a function of the system

size, where the vertical axis represents qL

scaled by H and the horizontal is s scaled

by sb or sΓ. The dotted line represents the

result for the Griffith criterion.

0 1 20

10

20

s/sbss/sb

qLH crackslip

sb

ss

FIG. 4. The shrinking rates at the

start of slip and at the first breaking are

plotted for the system size, as in Fig. 3.

FIG. 5. The two-dimensional

model. The blank triangles repre-

sent a crack.

Tm

nV

1

2

3

FIG. 6. A part of a random lattice

38

(a) C = 0.86 (b) C = 0.97

(c) C = 1.46 (d) C = 10.00FIG. 7. The time series of a crack pattern for L = 10

√2, µ = 1, τ = 0.01 and Cs = 0.5. The

black area represents cracks. The gray scale in (a),(b) and (d) indicates the energy densities of the

triangular cells of a lattice, and the dots in (c) indicate slipping elements.

39

0 1 2contraction ratio C/Cb

0

100

200

300

E

τ=0.1τ=0.01τ=0.001

FIG. 8. The change of the total energy

with contraction for L = 10√2, µ = 1,

Cs = 0.5 and τ = 0.001 for the solid line,

τ = 0.01 for the dotted line, and τ = 0.1

for the dot-dashed line.

FIG. 9. The time development of

cracks with fast relaxation. L = 10√2,

µ = 1, Cs = 0.5, τ = 0.001 and C = 1.54.

0 1 2contraction ratio C/Cb

0

1000

2000

τ=0.1τ=0.01τ=0.001

FIG. 10. The change of the total num-

ber of broken triangular cells for the sim-

ulations of Fig. 8.

FIG. 11. The time development of

cracks with slow relaxation. L = 10√2,

µ = 1, Cs = 0.5, τ = 0.2 and C = 1.09.

40

FIG. 12. A final crack pattern on a

sticky bottom. L = 20√2, µ = 1.0,

τ = 0.01, Cs = 1.0 and C = 10.

FIG. 13. A final crack pattern on a

slippery bottom. L = 20√2, µ = 1.0,

τ = 0.01, Cs = 0.1 and C = 10.

0 5 10contraction ratio C/Cb

0

1000

2000

3000

4000

5000

6000

Cs=1.0Cb

Cs=0.7Cb

Cs=0.5Cb

Cs=0.2Cb

Cs=0.1Cb

FIG. 14. The change of the total

number of broken triangular cells for

L = 20√2, µ = 1.0, τ = 0.01 and

Cs = 0.1, 0.2, 0.5, 0.7, 1.0.

0.0 0.2 0.4 0.6 0.8critical contraction ratio to slip Cs/Cb

0

1000

2000

3000

4000

5000

6000

FIG. 15. The final number of broken

triangles, where we plot the values at

C = 10 in Fig. 14 for Cs.

41

FIG. 16. A crack pattern for small µ.

L =10√2, µ = 0.02, Cs = 0.5, τ = 0.01

and C = 10.

0 1 2 3 4 5contraction ratio C/Cb

0

1000

2000

3000

µ=0.02κµ=0.1κµ=1.0κ

FIG. 17. The change of the total num-

ber of broken triangles for L = 10√2,

Cs = 0.5, τ = 0.01 and µ = 0.02, 0.1,

1.0.

FIG. 18. A crack pattern on a regular

triangular lattice for the same parameters

as in Fig. 7 (d).

42